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Properties

Label 29.2.e.a.4.1
Level $29$
Weight $2$
Character 29.4
Analytic conductor $0.232$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [29,2,Mod(4,29)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("29.4"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(29, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 29.e (of order \(14\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.231566165862\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{14})\)
Coefficient field: 12.0.7877952219361.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3x^{11} + 13x^{9} - 18x^{8} - 14x^{7} + 57x^{6} - 28x^{5} - 72x^{4} + 104x^{3} - 96x + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label 4.1
Root \(0.639551 + 1.26134i\) of defining polynomial
Character \(\chi\) \(=\) 29.4
Dual form 29.2.e.a.22.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.21089 - 0.504621i) q^{2} +(-1.23248 - 2.55926i) q^{3} +(2.83146 + 1.36356i) q^{4} +(0.0128801 - 0.0564316i) q^{5} +(1.43341 + 6.28018i) q^{6} +(1.40728 - 0.677709i) q^{7} +(-2.02596 - 1.61565i) q^{8} +(-3.16036 + 3.96297i) q^{9} +(-0.0569531 + 0.118264i) q^{10} +(3.10613 - 2.47705i) q^{11} -8.92699i q^{12} +(-0.252504 - 0.316631i) q^{13} +(-3.45332 + 0.788198i) q^{14} +(-0.160298 + 0.0365869i) q^{15} +(-0.254963 - 0.319714i) q^{16} +5.16843i q^{17} +(8.98701 - 7.16690i) q^{18} +(-1.49728 + 3.10914i) q^{19} +(0.113417 - 0.142221i) q^{20} +(-3.46887 - 2.76633i) q^{21} +(-8.11728 + 3.90908i) q^{22} +(-0.0512209 - 0.224414i) q^{23} +(-1.63793 + 7.17623i) q^{24} +(4.50183 + 2.16797i) q^{25} +(0.398481 + 0.827455i) q^{26} +(5.72931 + 1.30768i) q^{27} +4.90874 q^{28} +(5.00751 - 1.98113i) q^{29} +0.372863 q^{30} +(-4.21005 - 0.960917i) q^{31} +(2.65101 + 5.50488i) q^{32} +(-10.1677 - 4.89649i) q^{33} +(2.60810 - 11.4268i) q^{34} +(-0.0201183 - 0.0881438i) q^{35} +(-14.3522 + 6.91164i) q^{36} +(-4.19898 - 3.34857i) q^{37} +(4.87927 - 6.11841i) q^{38} +(-0.499135 + 1.03647i) q^{39} +(-0.117269 + 0.0935185i) q^{40} +1.46294i q^{41} +(6.27335 + 7.86653i) q^{42} +(6.32061 - 1.44264i) q^{43} +(12.1725 - 2.77829i) q^{44} +(0.182931 + 0.229388i) q^{45} +0.522001i q^{46} +(-7.48524 + 5.96928i) q^{47} +(-0.503996 + 1.04656i) q^{48} +(-2.84329 + 3.56537i) q^{49} +(-8.85904 - 7.06485i) q^{50} +(13.2274 - 6.36997i) q^{51} +(-0.283211 - 1.24083i) q^{52} +(0.398044 - 1.74395i) q^{53} +(-12.0070 - 5.78226i) q^{54} +(-0.0997767 - 0.207188i) q^{55} +(-3.94604 - 0.900657i) q^{56} +9.80247 q^{57} +(-12.0708 + 1.85317i) q^{58} -1.05216 q^{59} +(-0.503764 - 0.114981i) q^{60} +(1.38382 + 2.87352i) q^{61} +(8.82307 + 4.24897i) q^{62} +(-1.76177 + 7.71880i) q^{63} +(-2.90123 - 12.7111i) q^{64} +(-0.0211203 + 0.0101710i) q^{65} +(20.0087 + 15.9564i) q^{66} +(-6.77893 + 8.50052i) q^{67} +(-7.04745 + 14.6342i) q^{68} +(-0.511205 + 0.407672i) q^{69} +0.205028i q^{70} +(-8.38773 - 10.5179i) q^{71} +(12.8056 - 2.92279i) q^{72} +(6.83974 - 1.56113i) q^{73} +(7.59372 + 9.52223i) q^{74} -14.1933i q^{75} +(-8.47898 + 6.76176i) q^{76} +(2.69246 - 5.59095i) q^{77} +(1.62656 - 2.03964i) q^{78} +(3.74078 + 2.98318i) q^{79} +(-0.0213259 + 0.0102700i) q^{80} +(-0.330785 - 1.44926i) q^{81} +(0.738232 - 3.23440i) q^{82} +(-7.58927 - 3.65480i) q^{83} +(-6.04990 - 12.5628i) q^{84} +(0.291662 + 0.0665701i) q^{85} -14.7022 q^{86} +(-11.2419 - 10.3738i) q^{87} -10.2950 q^{88} +(1.10427 + 0.252043i) q^{89} +(-0.288686 - 0.599462i) q^{90} +(-0.569927 - 0.274462i) q^{91} +(0.160971 - 0.705260i) q^{92} +(2.72955 + 11.9589i) q^{93} +(19.5613 - 9.42022i) q^{94} +(0.156168 + 0.124540i) q^{95} +(10.8211 - 13.5693i) q^{96} +(7.46850 - 15.5085i) q^{97} +(8.08536 - 6.44786i) q^{98} +20.1379i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 7 q^{2} - 7 q^{3} - q^{4} - q^{5} - 3 q^{6} - 11 q^{7} + 14 q^{8} - 3 q^{9} - 7 q^{10} + 7 q^{11} + 9 q^{13} - 7 q^{14} + 7 q^{15} + 9 q^{16} + 42 q^{18} - 7 q^{19} - 11 q^{20} - 7 q^{21} - 4 q^{22}+ \cdots - 42 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/29\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{1}{14}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.21089 − 0.504621i −1.56334 − 0.356821i −0.648682 − 0.761060i \(-0.724679\pi\)
−0.914654 + 0.404239i \(0.867537\pi\)
\(3\) −1.23248 − 2.55926i −0.711571 − 1.47759i −0.871465 − 0.490457i \(-0.836830\pi\)
0.159895 − 0.987134i \(-0.448885\pi\)
\(4\) 2.83146 + 1.36356i 1.41573 + 0.681779i
\(5\) 0.0128801 − 0.0564316i 0.00576017 − 0.0252370i −0.971966 − 0.235121i \(-0.924451\pi\)
0.977726 + 0.209884i \(0.0673086\pi\)
\(6\) 1.43341 + 6.28018i 0.585188 + 2.56387i
\(7\) 1.40728 − 0.677709i 0.531901 − 0.256150i −0.148600 − 0.988897i \(-0.547477\pi\)
0.680501 + 0.732747i \(0.261762\pi\)
\(8\) −2.02596 − 1.61565i −0.716287 − 0.571220i
\(9\) −3.16036 + 3.96297i −1.05345 + 1.32099i
\(10\) −0.0569531 + 0.118264i −0.0180102 + 0.0373985i
\(11\) 3.10613 − 2.47705i 0.936533 − 0.746860i −0.0310229 − 0.999519i \(-0.509876\pi\)
0.967555 + 0.252659i \(0.0813051\pi\)
\(12\) − 8.92699i − 2.57700i
\(13\) −0.252504 − 0.316631i −0.0700321 − 0.0878175i 0.745581 − 0.666415i \(-0.232172\pi\)
−0.815614 + 0.578597i \(0.803600\pi\)
\(14\) −3.45332 + 0.788198i −0.922939 + 0.210655i
\(15\) −0.160298 + 0.0365869i −0.0413887 + 0.00944670i
\(16\) −0.254963 − 0.319714i −0.0637408 − 0.0799285i
\(17\) 5.16843i 1.25353i 0.779209 + 0.626764i \(0.215621\pi\)
−0.779209 + 0.626764i \(0.784379\pi\)
\(18\) 8.98701 − 7.16690i 2.11826 − 1.68926i
\(19\) −1.49728 + 3.10914i −0.343500 + 0.713286i −0.999126 − 0.0418005i \(-0.986691\pi\)
0.655626 + 0.755086i \(0.272405\pi\)
\(20\) 0.113417 − 0.142221i 0.0253609 − 0.0318015i
\(21\) −3.46887 − 2.76633i −0.756970 − 0.603663i
\(22\) −8.11728 + 3.90908i −1.73061 + 0.833418i
\(23\) −0.0512209 − 0.224414i −0.0106803 − 0.0467935i 0.969307 − 0.245854i \(-0.0790682\pi\)
−0.979987 + 0.199060i \(0.936211\pi\)
\(24\) −1.63793 + 7.17623i −0.334341 + 1.46484i
\(25\) 4.50183 + 2.16797i 0.900365 + 0.433593i
\(26\) 0.398481 + 0.827455i 0.0781486 + 0.162277i
\(27\) 5.72931 + 1.30768i 1.10261 + 0.251663i
\(28\) 4.90874 0.927664
\(29\) 5.00751 − 1.98113i 0.929870 − 0.367887i
\(30\) 0.372863 0.0680752
\(31\) −4.21005 − 0.960917i −0.756148 − 0.172586i −0.172967 − 0.984928i \(-0.555335\pi\)
−0.583182 + 0.812342i \(0.698192\pi\)
\(32\) 2.65101 + 5.50488i 0.468637 + 0.973135i
\(33\) −10.1677 − 4.89649i −1.76996 − 0.852369i
\(34\) 2.60810 − 11.4268i 0.447285 − 1.95968i
\(35\) −0.0201183 − 0.0881438i −0.00340061 − 0.0148990i
\(36\) −14.3522 + 6.91164i −2.39203 + 1.15194i
\(37\) −4.19898 − 3.34857i −0.690308 − 0.550502i 0.214289 − 0.976770i \(-0.431257\pi\)
−0.904597 + 0.426268i \(0.859828\pi\)
\(38\) 4.87927 − 6.11841i 0.791521 − 0.992536i
\(39\) −0.499135 + 1.03647i −0.0799256 + 0.165967i
\(40\) −0.117269 + 0.0935185i −0.0185418 + 0.0147866i
\(41\) 1.46294i 0.228473i 0.993454 + 0.114237i \(0.0364422\pi\)
−0.993454 + 0.114237i \(0.963558\pi\)
\(42\) 6.27335 + 7.86653i 0.967998 + 1.21383i
\(43\) 6.32061 − 1.44264i 0.963884 − 0.220000i 0.288511 − 0.957477i \(-0.406840\pi\)
0.675373 + 0.737476i \(0.263983\pi\)
\(44\) 12.1725 − 2.77829i 1.83507 − 0.418842i
\(45\) 0.182931 + 0.229388i 0.0272697 + 0.0341951i
\(46\) 0.522001i 0.0769648i
\(47\) −7.48524 + 5.96928i −1.09184 + 0.870709i −0.992242 − 0.124319i \(-0.960325\pi\)
−0.0995928 + 0.995028i \(0.531754\pi\)
\(48\) −0.503996 + 1.04656i −0.0727455 + 0.151058i
\(49\) −2.84329 + 3.56537i −0.406184 + 0.509339i
\(50\) −8.85904 − 7.06485i −1.25286 − 0.999121i
\(51\) 13.2274 − 6.36997i 1.85220 − 0.891974i
\(52\) −0.283211 − 1.24083i −0.0392743 − 0.172072i
\(53\) 0.398044 − 1.74395i 0.0546756 − 0.239549i −0.940205 − 0.340610i \(-0.889366\pi\)
0.994880 + 0.101061i \(0.0322236\pi\)
\(54\) −12.0070 − 5.78226i −1.63394 − 0.786866i
\(55\) −0.0997767 − 0.207188i −0.0134539 − 0.0279373i
\(56\) −3.94604 − 0.900657i −0.527311 − 0.120355i
\(57\) 9.80247 1.29837
\(58\) −12.0708 + 1.85317i −1.58497 + 0.243333i
\(59\) −1.05216 −0.136980 −0.0684900 − 0.997652i \(-0.521818\pi\)
−0.0684900 + 0.997652i \(0.521818\pi\)
\(60\) −0.503764 − 0.114981i −0.0650357 − 0.0148440i
\(61\) 1.38382 + 2.87352i 0.177179 + 0.367917i 0.970578 − 0.240785i \(-0.0774049\pi\)
−0.793399 + 0.608702i \(0.791691\pi\)
\(62\) 8.82307 + 4.24897i 1.12053 + 0.539619i
\(63\) −1.76177 + 7.71880i −0.221962 + 0.972478i
\(64\) −2.90123 − 12.7111i −0.362653 − 1.58889i
\(65\) −0.0211203 + 0.0101710i −0.00261964 + 0.00126155i
\(66\) 20.0087 + 15.9564i 2.46290 + 1.96410i
\(67\) −6.77893 + 8.50052i −0.828179 + 1.03850i 0.170409 + 0.985373i \(0.445491\pi\)
−0.998588 + 0.0531298i \(0.983080\pi\)
\(68\) −7.04745 + 14.6342i −0.854628 + 1.77465i
\(69\) −0.511205 + 0.407672i −0.0615418 + 0.0490780i
\(70\) 0.205028i 0.0245056i
\(71\) −8.38773 − 10.5179i −0.995440 − 1.24824i −0.968606 − 0.248603i \(-0.920029\pi\)
−0.0268346 − 0.999640i \(-0.508543\pi\)
\(72\) 12.8056 − 2.92279i 1.50915 − 0.344454i
\(73\) 6.83974 − 1.56113i 0.800531 − 0.182716i 0.197363 − 0.980330i \(-0.436762\pi\)
0.603167 + 0.797615i \(0.293905\pi\)
\(74\) 7.59372 + 9.52223i 0.882752 + 1.10694i
\(75\) − 14.1933i − 1.63890i
\(76\) −8.47898 + 6.76176i −0.972605 + 0.775627i
\(77\) 2.69246 − 5.59095i 0.306834 − 0.637148i
\(78\) 1.62656 − 2.03964i 0.184171 − 0.230943i
\(79\) 3.74078 + 2.98318i 0.420871 + 0.335634i 0.810916 − 0.585162i \(-0.198969\pi\)
−0.390045 + 0.920796i \(0.627541\pi\)
\(80\) −0.0213259 + 0.0102700i −0.00238431 + 0.00114822i
\(81\) −0.330785 − 1.44926i −0.0367539 − 0.161029i
\(82\) 0.738232 − 3.23440i 0.0815241 − 0.357180i
\(83\) −7.58927 − 3.65480i −0.833030 − 0.401166i −0.0317796 − 0.999495i \(-0.510117\pi\)
−0.801251 + 0.598329i \(0.795832\pi\)
\(84\) −6.04990 − 12.5628i −0.660099 − 1.37071i
\(85\) 0.291662 + 0.0665701i 0.0316352 + 0.00722054i
\(86\) −14.7022 −1.58537
\(87\) −11.2419 − 10.3738i −1.20526 − 1.11219i
\(88\) −10.2950 −1.09745
\(89\) 1.10427 + 0.252043i 0.117053 + 0.0267165i 0.280646 − 0.959811i \(-0.409451\pi\)
−0.163593 + 0.986528i \(0.552308\pi\)
\(90\) −0.288686 − 0.599462i −0.0304301 − 0.0631888i
\(91\) −0.569927 − 0.274462i −0.0597446 − 0.0287715i
\(92\) 0.160971 − 0.705260i 0.0167824 − 0.0735284i
\(93\) 2.72955 + 11.9589i 0.283041 + 1.24009i
\(94\) 19.5613 − 9.42022i 2.01759 − 0.971621i
\(95\) 0.156168 + 0.124540i 0.0160225 + 0.0127775i
\(96\) 10.8211 − 13.5693i 1.10443 − 1.38491i
\(97\) 7.46850 − 15.5085i 0.758311 − 1.57465i −0.0588689 − 0.998266i \(-0.518749\pi\)
0.817180 − 0.576383i \(-0.195536\pi\)
\(98\) 8.08536 − 6.44786i 0.816745 − 0.651332i
\(99\) 20.1379i 2.02393i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 29.2.e.a.4.1 ✓ 12
3.2 odd 2 261.2.o.a.91.2 12
4.3 odd 2 464.2.y.d.33.2 12
5.2 odd 4 725.2.p.a.149.4 24
5.3 odd 4 725.2.p.a.149.1 24
5.4 even 2 725.2.q.a.526.2 12
29.2 odd 28 841.2.d.l.190.1 24
29.3 odd 28 841.2.d.m.645.1 24
29.4 even 14 841.2.e.f.267.1 12
29.5 even 14 841.2.e.a.651.1 12
29.6 even 14 841.2.b.e.840.2 12
29.7 even 7 841.2.e.i.196.2 12
29.8 odd 28 841.2.d.k.778.1 24
29.9 even 14 841.2.e.e.63.2 12
29.10 odd 28 841.2.d.k.574.1 24
29.11 odd 28 841.2.d.l.571.1 24
29.12 odd 4 841.2.d.m.605.4 24
29.13 even 14 841.2.e.h.270.2 12
29.14 odd 28 841.2.a.k.1.11 12
29.15 odd 28 841.2.a.k.1.2 12
29.16 even 7 841.2.e.a.270.1 12
29.17 odd 4 841.2.d.m.605.1 24
29.18 odd 28 841.2.d.l.571.4 24
29.19 odd 28 841.2.d.k.574.4 24
29.20 even 7 841.2.e.f.63.1 12
29.21 odd 28 841.2.d.k.778.4 24
29.22 even 14 inner 29.2.e.a.22.1 yes 12
29.23 even 7 841.2.b.e.840.11 12
29.24 even 7 841.2.e.h.651.2 12
29.25 even 7 841.2.e.e.267.2 12
29.26 odd 28 841.2.d.m.645.4 24
29.27 odd 28 841.2.d.l.190.4 24
29.28 even 2 841.2.e.i.236.2 12
87.14 even 28 7569.2.a.bp.1.2 12
87.44 even 28 7569.2.a.bp.1.11 12
87.80 odd 14 261.2.o.a.109.2 12
116.51 odd 14 464.2.y.d.225.2 12
145.22 odd 28 725.2.p.a.399.1 24
145.109 even 14 725.2.q.a.51.2 12
145.138 odd 28 725.2.p.a.399.4 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
29.2.e.a.4.1 ✓ 12 1.1 even 1 trivial
29.2.e.a.22.1 yes 12 29.22 even 14 inner
261.2.o.a.91.2 12 3.2 odd 2
261.2.o.a.109.2 12 87.80 odd 14
464.2.y.d.33.2 12 4.3 odd 2
464.2.y.d.225.2 12 116.51 odd 14
725.2.p.a.149.1 24 5.3 odd 4
725.2.p.a.149.4 24 5.2 odd 4
725.2.p.a.399.1 24 145.22 odd 28
725.2.p.a.399.4 24 145.138 odd 28
725.2.q.a.51.2 12 145.109 even 14
725.2.q.a.526.2 12 5.4 even 2
841.2.a.k.1.2 12 29.15 odd 28
841.2.a.k.1.11 12 29.14 odd 28
841.2.b.e.840.2 12 29.6 even 14
841.2.b.e.840.11 12 29.23 even 7
841.2.d.k.574.1 24 29.10 odd 28
841.2.d.k.574.4 24 29.19 odd 28
841.2.d.k.778.1 24 29.8 odd 28
841.2.d.k.778.4 24 29.21 odd 28
841.2.d.l.190.1 24 29.2 odd 28
841.2.d.l.190.4 24 29.27 odd 28
841.2.d.l.571.1 24 29.11 odd 28
841.2.d.l.571.4 24 29.18 odd 28
841.2.d.m.605.1 24 29.17 odd 4
841.2.d.m.605.4 24 29.12 odd 4
841.2.d.m.645.1 24 29.3 odd 28
841.2.d.m.645.4 24 29.26 odd 28
841.2.e.a.270.1 12 29.16 even 7
841.2.e.a.651.1 12 29.5 even 14
841.2.e.e.63.2 12 29.9 even 14
841.2.e.e.267.2 12 29.25 even 7
841.2.e.f.63.1 12 29.20 even 7
841.2.e.f.267.1 12 29.4 even 14
841.2.e.h.270.2 12 29.13 even 14
841.2.e.h.651.2 12 29.24 even 7
841.2.e.i.196.2 12 29.7 even 7
841.2.e.i.236.2 12 29.28 even 2
7569.2.a.bp.1.2 12 87.14 even 28
7569.2.a.bp.1.11 12 87.44 even 28